Simplify each expression. Assume all variables represent nonzero real numbers. (p⁴/q)²

Verified step by step guidance

1. Start with the given expression: \(\left( \frac{p^{4}}{q} \right)^{2}\).
2. Apply the power of a quotient rule, which states that \(\left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}}\). So rewrite the expression as \(\frac{\left(p^{4}\right)^{2}}{q^{2}}\).
3. Next, apply the power of a power rule to the numerator: \(\left(p^{4}\right)^{2} = p^{4 \times 2} = p^{8}\).
4. Now the expression is simplified to \(\frac{p^{8}}{q^{2}}\).
5. Since all variables represent nonzero real numbers, this is the simplified form of the expression.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include (a^m)^n = a^(m*n) and (a/b)^n = a^n / b^n. Applying these rules allows you to simplify expressions like (p^4 / q)^2 by raising both numerator and denominator to the power of 2.

Simplifying Rational Expressions

A rational expression is a fraction where the numerator and/or denominator contain variables with exponents. Simplifying involves applying exponent rules and reducing the expression to its simplest form, assuming variables are nonzero to avoid division by zero.

Assumptions About Variables

When simplifying expressions with variables, it is important to note any assumptions, such as variables representing nonzero real numbers. This ensures operations like division are valid and prevents undefined expressions.